# Control Barrier Functions for Systems with High Relative Degree

arXiv: https://arxiv.org/abs/1903.04706 · HTML: https://mlsanity.com/paper/1903.04706/

## Key facts

- Cited by 31 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 12 Mar 2019.

## Built on by (most cited first, 30 of 31)

- [SafeDiffuser: Safe Planning with Diffusion Probabilistic Models](https://mlsanity.com/paper/2306.00148/) (2023)
- [Drive Anywhere: Generalizable End-to-end Autonomous Driving with Multi-modal Foundation Models](https://mlsanity.com/paper/2310.17642/) (2023)
- [Safe Control Under Input Limits with Neural Control Barrier Functions](https://mlsanity.com/paper/2211.11056/) (2022)
- [CGD: Constraint-Guided Diffusion Policies for UAV Trajectory Planning](https://mlsanity.com/paper/2405.01758/) (2024)
- [Learning Robust Output Control Barrier Functions from Safe Expert Demonstrations](https://mlsanity.com/paper/2111.09971/) (2021)
- [Exact Verification of ReLU Neural Control Barrier Functions](https://mlsanity.com/paper/2310.09360/) (2023)
- [Verification of Neural Control Barrier Functions with Symbolic Derivative Bounds Propagation](https://mlsanity.com/paper/2410.16281/) (2024)
- [On the Forward Invariance of Neural ODEs](https://mlsanity.com/paper/2210.04763/) (2022)
- [Online Control Barrier Functions for Decentralized Multi-Agent Navigation](https://mlsanity.com/paper/2303.04313/) (2023)
- [Differentiable Control Barrier Functions for Vision-based End-to-End Autonomous Driving](https://mlsanity.com/paper/2203.02401/) (2022)
- [Sequential Neural Barriers for Scalable Dynamic Obstacle Avoidance](https://mlsanity.com/paper/2307.03015/) (2023)
- [Differentiable Safe Controller Design through Control Barrier Functions](https://mlsanity.com/paper/2209.10034/) (2022)
- [Discrete GCBF Proximal Policy Optimization for Multi-agent Safe Optimal Control](https://mlsanity.com/paper/2502.03640/) (2025)
- [Differentiable Predictive Control with Safety Guarantees: A Control Barrier Function Approach](https://mlsanity.com/paper/2208.02319/) (2022)
- [Learning Robust and Correct Controllers from Signal Temporal Logic Specifications Using BarrierNet](https://mlsanity.com/paper/2304.06160/) (2023)
- [Learning Stability Attention in Vision-based End-to-end Driving Policies](https://mlsanity.com/paper/2304.02733/) (2023)
- [Dynamic High-Order Control Barrier Functions with Diffuser for Safety-Critical Trajectory Planning at Signal-Free Intersections](https://mlsanity.com/paper/2412.00162/) (2024)
- [See Less, Drive Better: Generalizable End-to-End Autonomous Driving via Foundation Models Stochastic Patch Selection](https://mlsanity.com/paper/2601.10707/) (2026)
- [Safe Neural Control for Non-Affine Control Systems with Differentiable Control Barrier Functions](https://mlsanity.com/paper/2309.04492/) (2023)
- [Learning Adaptive Safety for Multi-Agent Systems](https://mlsanity.com/paper/2309.10657/) (2023)
- [Agile Decision-Making and Safety-Critical Motion Planning for Emergency Autonomous Vehicles](https://mlsanity.com/paper/2409.08665/) (2024)
- [Gaussian Control Barrier Functions : A Non-Parametric Paradigm to Safety](https://mlsanity.com/paper/2203.15474/) (2022)
- [Learning Feasibility Constraints for Control Barrier Functions](https://mlsanity.com/paper/2303.09403/) (2023)
- [Trust-Aware Resilient Control and Coordination of Connected and Automated Vehicles](https://mlsanity.com/paper/2305.16818/) (2023)
- [Robust AI Driving Strategy for Autonomous Vehicles](https://mlsanity.com/paper/2207.07829/) (2022)
- [The Feasibility Theory of Constrained Reinforcement Learning: A Tutorial Study](https://mlsanity.com/paper/2404.10064/) (2024)
- [Permissive Safety Through Trusted Inference: Verifiable Belief-Space Neural Safety Filters for Assured Interactive Robotics](https://mlsanity.com/paper/2606.02562/) (2026)
- [Optimal Parameter Adaptation for Safety-Critical Control via Safe Barrier Bayesian Optimization](https://mlsanity.com/paper/2503.19349/) (2025)
- [Neural Network-assisted Interval Reachability for Systems with Control Barrier Function-Based Safe Controllers](https://mlsanity.com/paper/2504.08249/) (2025)
- [End-to-End Learning of Safe Optimal Feedback Control in High Dimensions with Control Barrier Function Layers](https://mlsanity.com/paper/2607.20674/) (2026)

## Abstract (arXiv)

This paper extends control barrier functions (CBFs) to high order control barrier functions (HOCBFs) that can be used for high relative degree constraints. The proposed HOCBFs are more general than recently proposed (exponential) HOCBFs. We introduce high order barrier functions (HOBF), and show that their satisfaction of Lyapunov-like conditions implies the forward invariance of the intersection of a series of sets. We then introduce HOCBF, and show that any control input that satisfies the HOCBF constraints renders the intersection of a series of sets forward invariant. We formulate optimal control problems with constraints given by HOCBF and control Lyapunov functions (CLF) and analyze the influence of the choice of the class K functions used in the definition of the HOCBF on the size of the feasible control region. We also provide a promising method to address the conflict between HOCBF constraints and control limitations by penalizing the class K functions. We illustrate the proposed method on an adaptive cruise control problem.